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This is the appendix of the paper "Name-passing calculi: from fusions to preorders and types" (D Hirschkoff, JM. Madiot, D. Sangiorgi), to appear in LICS'2013. APPENDIX A. Reduction-closed barbed congruence (Section II) Definition 28 (Reduction-closed barbed congruence). Let L be a process calculus, in which a reduction relation −→ L and barb predicates ↓ L a , for each a in a given set of names, have been defined.Then reduction-closed barbed congruence in L, written L , is the largest symmetric relation on the processes of L that is context-closed, reduction-closed, and barb-preserving.B. Proofs of impossibility results (Section III) Statement of Theorem 3: A typed calculus with fusions that is plain and supports narrowing has trivial subtyping.Proof Sketch: We define the following active context:Note that in E we only use b as an output object. The intention is that, given some process P , and u, v, c some fresh names, E[P ] should reduce to P {a/b}. Indeed, by applying hypothesis (2) twice, we haveSuppose U ≤ T , we show Γ, a : T P iff Γ, a : U P . The implication from left to right is narrowing. To prove the right to left implication, suppose Γ, a : U P , and prove Γ, a : T P . By injective name substitution we have Γ, b : U P {b/a} for some fresh b.In the typing environment Γ, b:U, u: T, v: T, c:T, a:T the process ub is well-typed thanks to narrowing and weakening, hence so is (ub | uc | va | vc | P {b/a}). By the restriction rule we get Γ, a:T, u: T, v: T E[P {b/a}], the latter reducing to P {b/a}{a/b} by (4). Since b has been taken fresh, P {b/a}{a/b} = P . Hence, by Subject Reduction, Γ, a:T, u: T, v: T P . We finally deduce Γ, a : T P by Strengthening. Statement of Theorem 4: Suppose a typed calculus with fusions is plain and there is at least one prefix α with object b, different from the subject, and there are two types S and T such that S ≤ T and one of the following forms of narrowing holds for all Γ:1) whenever Γ, b : T α. 0, we also have Γ, b : S α. 0; 2) whenever Γ, b : S α. 0, we also have Γ, b : T α. 0. Then S and T are interchangeable in all typing judgements.Proof Sketch: For all ∆ we prove that ∆, x : T P iff ∆, x : S P . Let x 1 , x 2 , a 1 and a 2 be fresh names.We will prove that ∆ i P {x 1 /x} implies ∆ i P {x 2 /x} for all i ∈ {1, 2}. From there it is enough to conclude using weakening, strengthening and injective substitutions. We use D = a 1 x 1 | a 2 x 2 | a 1 y | a 2 y to simulate a substitution:We have to prove that ∆ = ∆ i , a 1 : T a1 , a 2 : T a2 , y : T y D for some types T a1 T a2 , T y . We note a the subject of α. Using the plainness of the subtyping, we can suppose that a is any of a 1 or a 2 and that b is any of x 1 , x 2 or y, so to apply the hypothesis on different cases. There are eight subcases, along the cases from the hypothesis, i, and the form of α.• (1), i = 1, α = a 2 x 2 : T a1 = T a2 = T , T y = T ; • (1), i = 1, α = a 1 y: T a1 = T , T a2 = S, T y = S; • (2), i = 1, α = a 1 x 1 : T a1 = T a2 = S, T y = S; • (2), i = 1, α = a 2 y : T a1 = T , T a2 = S, T y = T ...

This is the appendix of the paper "Name-passing calculi: from fusions to preorders and types" (D Hirschkoff, JM. Madiot, D. Sangiorgi), to appear in LICS'2013. APPENDIX A. Reduction-closed barbed congruence (Section II) Definition 28 (Reduction-closed barbed congruence). Let L be a process calculus, in which a reduction relation −→ L and barb predicates ↓ L a , for each a in a given set of names, have been defined.Then reduction-closed barbed congruence in L, written L , is the largest symmetric relation on the processes of L that is context-closed, reduction-closed, and barb-preserving.B. Proofs of impossibility results (Section III) Statement of Theorem 3: A typed calculus with fusions that is plain and supports narrowing has trivial subtyping.Proof Sketch: We define the following active context:Note that in E we only use b as an output object. The intention is that, given some process P , and u, v, c some fresh names, E[P ] should reduce to P {a/b}. Indeed, by applying hypothesis (2) twice, we haveSuppose U ≤ T , we show Γ, a : T P iff Γ, a : U P . The implication from left to right is narrowing. To prove the right to left implication, suppose Γ, a : U P , and prove Γ, a : T P . By injective name substitution we have Γ, b : U P {b/a} for some fresh b.In the typing environment Γ, b:U, u: T, v: T, c:T, a:T the process ub is well-typed thanks to narrowing and weakening, hence so is (ub | uc | va | vc | P {b/a}). By the restriction rule we get Γ, a:T, u: T, v: T E[P {b/a}], the latter reducing to P {b/a}{a/b} by (4). Since b has been taken fresh, P {b/a}{a/b} = P . Hence, by Subject Reduction, Γ, a:T, u: T, v: T P . We finally deduce Γ, a : T P by Strengthening. Statement of Theorem 4: Suppose a typed calculus with fusions is plain and there is at least one prefix α with object b, different from the subject, and there are two types S and T such that S ≤ T and one of the following forms of narrowing holds for all Γ:1) whenever Γ, b : T α. 0, we also have Γ, b : S α. 0; 2) whenever Γ, b : S α. 0, we also have Γ, b : T α. 0. Then S and T are interchangeable in all typing judgements.Proof Sketch: For all ∆ we prove that ∆, x : T P iff ∆, x : S P . Let x 1 , x 2 , a 1 and a 2 be fresh names.We will prove that ∆ i P {x 1 /x} implies ∆ i P {x 2 /x} for all i ∈ {1, 2}. From there it is enough to conclude using weakening, strengthening and injective substitutions. We use D = a 1 x 1 | a 2 x 2 | a 1 y | a 2 y to simulate a substitution:We have to prove that ∆ = ∆ i , a 1 : T a1 , a 2 : T a2 , y : T y D for some types T a1 T a2 , T y . We note a the subject of α. Using the plainness of the subtyping, we can suppose that a is any of a 1 or a 2 and that b is any of x 1 , x 2 or y, so to apply the hypothesis on different cases. There are eight subcases, along the cases from the hypothesis, i, and the form of α.• (1), i = 1, α = a 2 x 2 : T a1 = T a2 = T , T y = T ; • (1), i = 1, α = a 1 y: T a1 = T , T a2 = S, T y = S; • (2), i = 1, α = a 1 x 1 : T a1 = T a2 = S, T y = S; • (2), i = 1, α = a 2 y : T a1 = T , T a2 = S, T y = T ...